Solve for x e^(3-x)=(e^3)^(-x)

$e^{3 - x} = {(e^{3})}^{- x}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$e^{3 - x} = e^{3 (- x)}$

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 - x = 3 (- x)$

Solve for $x$ .

Multiply $- 1$ by $3$ .

$3 - x = - 3 x$

Move all terms containing $x$ to the left side of the equation.

Add $3 x$ to both sides of the equation.

$3 - x + 3 x = 0$

Add $- x$ and $3 x$ .

$3 + 2 x = 0$

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Divide each term in $2 x = - 3$ by $2$ and simplify.

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

$x = - 1 \frac{1}{2}$